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Register a new userLocal times for continuous paths of arbitrary regularity
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Donghan Kim
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We study a continuous pathwise local time of order p for continuous functions with finite p-th variation along a sequence of time partitions, for even integers p >= 2. With this notion, we establish a Tanaka-type change of variable formula, as well as Tanaka-Meyer formulae. We also derive some identities involving this high-order pathwise local time, each of which generalizes a corresponding identity from semimartingale theory. We then use collision local times between multiple functions of arbitrary regularity, to study the dynamics of ranked continuous functions of arbitrary regularity. We present also another definition of pathwise local time which is more natural for fractional Brownian Motions, and give a connection with the previous notion of local time.
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Let $B^{alpha_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${alpha_i}$ ($i=1,2$), and let $B^{alpha_1}$ and $B^{alpha_2}$ be independent. We prove that, if $frac{N_1}{alpha_1}+frac{N_2}{alpha_2}>d$, then the intersection local times of $B^{alpha_1}$ and $B^{alpha_2}$ exist, and have a continuous version. We also establish H{o}lder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points. One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre ({it J. Theor. Probab.} {bf 20} (2007)), where the existence of the intersection local times of two independent $(1,d)$-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.
We investigate the space-time regularity of the local time associated to Volterra-Levy processes, including Volterra processes driven by $alpha$-stable processes for $alphain(0,2]$. We show that the spatial regularity of the local time for Volterra-Levy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbaPtion of ODEs by a Volterra-Levy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differentiablility of the flow associated to such equations.
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann-Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann-Liouville process.
We consider sequences of additive functionals of difference approximations for uniformly non-degenerate multidimensional diffusions. The conditions are given, sufficient for such a sequence to converge weakly to a W-functional of the limiting process. The class of the W-functionals, that can be obtained as the limiting ones, is completely described in the terms of the associated W-measures, and coincides with the class of the functionals that are regular w.r.t. the phase variable.
We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the $M1$ topology and fails to hold in the $J1$ topology.
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